Mdthbs

It is safe to say that every mathematician, at some point in his career, has had some form of exposure to analysis. Quite often, it appears first in the form of an undergraduate course in real analysis. It is there that one is often exposed to a rigorous viewpoint to the techniques of calculus that one is already familiar with. At this stage, one might argue that real analysis is the study of real numbers, but is it? A big chunk of it involves algebraic properties, and as such lies in the realm of algebra. It is the order properties, though, that do have a sort of analysis point of view. Sure, some of these aspects generalise to the level of topologies, but not all. Completeness, for one, is clearly something that is central to analysis.

Similar arguments can be made for complex analysis and functional analysis.

Now, the question is: As for all the topics that are bunched together as analysis, is there any central theme to them? What topics would you say that belongs to this theme? And what are the underlying themes in these individual subtopics?

Add. It may be a subjective question, but having a rough idea of what the central themes of a certain field are helps one to construct appropriate questions. As such, I think it is important. I am not expecting a single answer, but more of a diverse set of opinions on the matter.

I think that I'd say that one of the underlying themes of analysis is, really, limits. In pretty much every subfield of analysis, we spend a lot of time trying to control the size of certain quantities, with taking limits in mind. This is especially true in PDEs, when we consistently desire norm estimates on various quantities. Let's just discuss the "basics" of the "basic" subjects (standard topics in real, complex, measure theory, functional). I'm going to keep this discussion loose, since we could quickly get into a very drawn-out and detailed discussion.

Real analysis is built on limits. Continuity, differentiability, integration, series, etc. all require the concept of limits. Complex analysis has a bit of a different "flavor" than the other core types, but it still requires limits to do, pretty much everything. By Goursat's theorem, holomorphicity is equivalent to complex differentiability over a neighborhood- limits. Integration and residue theory- limits. We can continue this (Laurent series, normal families, etc.). Lebesgue integration theory and the powerful theorems that come with it are all centered around, essentially, swapping limits, and much of measure theory is built around this. In functional analysis, we certainly have times where we have non-Hausdorff topologies, but limits are still central. Hilbert, Banach, and Frechet spaces all make use of a metric. We have things like uniform boundedness, compact operators, spectral theory, semigroups, Fourier analysis (this is a field in its own right, of course, but it deals with a lot of functional analysis), and much more, all of which deal with limits (either explicitly or via objects related to previously-discussed material). A significant subfield of analysis is PDEs. As I said earlier, PDEs often deals with obtaining proper norm estimates on certain quantities in appropriate function spaces to prove e.g. existence and regularity of solutions, once again highly dependent on limit arguments (and, of course, the norms themselves are limit-dependent).

Something else that I didn't touch on, but is important to discuss is just how many modes of convergence there are. Some common types of convergence of sequences of functions and operators are pointwise convergence, uniform convergence, local uniform convergence, almost everywhere convergence, convergence in measure, $L^p$ convergence, (more generally) convergence in norm, weak convergence, weak star convergence, uniform operator convergence, strong operator convergence, weak operator convergence, etc. I didn't distinguish between convergence for operators and functions too much here, but it is important to do so e.g. weak star convergence is pointwise convergence for elements of the dual, but I listed them as separate.

EDIT: The OP asked for some details. Of course, writing everything above in details would amount to me writing books! Instead of talking about everything, I'd like to talk about one pervasive concept in analysis that comes from limits- the integral. I'd like to note that much of the post deals with limits in many other ways as well, explicitly or otherwise. In real analysis, there are various equivalent ways that the integral is defined, but I'd like to use Riemann sums here: we say that a function is Riemann integrable on an interval $[a,b]$ if and only if there exists $Iin mathbb{R}$ so that

$$I=lim_{|P|rightarrow 0}sumlimits_{i=1}^n f(t_i)(x_i-x_{i-1}),$$

where $|P|$ denotes the size of the partition, where $t_iin [x_{i-1},x_i].$ We call $I$ the integral, and we denote it as $$I=intlimits_a^b f(s)ds.$$ The integral of a continuous (limits!) function is related to the derivative (limits!) through the fundamental theorem of calculus:

For $fin Cleft([a,b]right),$ a function $F$ satisfies $$F(x)-F(a)=intlimits_a^x f(s)ds$$ for any $xin [a,b]$ if and only if $F'=f$.

As the name of the theorem states, this is pretty important. All of this generalizes appropriately to higher dimensions, but I won't discuss that here. Sometimes, integrating a function on its own can be hard, so we approximate it with easier functions (or sometimes, we have a sequence of functions tending to something, and we want to know about the limit and how it integrates). A major theorem in an introductory real analysis class is that if we have a sequence of Riemann integrable functions $(f_n)$ converging uniformly on $[a,b]$ to $f$, then $f$ is Riemann integrable, and we can swap the limit and integral. So, we can swap these two limits. We can do the same for a series of functions that converges uniformly.

Okay, let's move on. In complex analysis, the integral is still of importance. Integrals in the complex plane are path integrals, which can be defined similarly. Complex analysis is centered on studying holomorphic functions, and a theorem of Morera relates this to the integral

Let $g:Omegarightarrowmathbb{C}$ be continuous, and $$intlimits_gamma g(z)dz=0$$ whenever $gamma=partial R$ and $RsubsetOmega$ is a rectangle (with sides parallel to the real and imaginary axes). Then, $g$ is holomorphic.

The integral pops up in many other fundamental ways here. One is in the form of the mean-value property, which states that $$f(z_0)=frac{1}{2pi}intlimits_0^{2pi} f(z_0+re^{itheta})dtheta$$ whenever $f$ is holomorphic on $Omega$ open and the closed disk centered $z_0$ of radius $r$ is contained in $Omega.$ We use the integral to prove other important theorems, such as the maximum modulus principle, Liouville's theorem, etc. We also use it to define a branch of the complex logarithm, to define the coefficients of a Laurent series, and to count zeros and poles of functions (argument principle). We also like to calculate various types of integrals in the complex plane where the integrand has singularities (often as a trick to calculate real integrals, which is especially relevant for calculating Fourier transforms). This uses the residue theorem, and residues are also calculated via taking limits. The theorem states that
$$intlimits_{partialOmega}f(z)dz=2pi isum_jtext{Res}_{z_j}(f),$$ where $f$ is holomorphic on an open set $Omega$, except at singularities ${z_j},$ each of which has a relatively compact neighborhood on which $f$ has a Laurent series (the residue is the $(-1)$'th indexed coefficient, which are also integrals by construction of the Laurent series). I think that's enough about complex analysis.

Now, let's talk a bit about measure theory. The Riemann integral is somewhat restrictive, so we generalize it to the Lebesgue integral (I have a post about the construction, see How to calculate an integral given a measure?). Note the involvement of limits in the post. If a function is Riemann integrable, then it is equivalent to its Lebesgue integral. We can define the Lebesgue integral on any measure space. Two of the biggest theorems are the monotone and dominated convergence theorems:

If $f_jin L^1(X,mu)$, $0leq f_1(x)leq f_2(x)leq cdots,$ and $|f_j|_{L^1}leq C<infty,$ then $lim_j f_j(x)=f(x),$ with $fin L^1(X,mu),$ and $|f_j-f|_{L^1}rightarrow 0.$

and

If $f_jin L^1(X,mu)$ and $lim_j f_j(x)=f(x)$ $mu$-a.e., and there is an $Fin L^1(X,mu)$ so that $F$ dominates each $|f_j|$ pointwise $mu$-a.e., then $fin L^1(X,mu)$ and $|f-j-f|_{L^1}rightarrow 0

We have immediate generalization to $L^p$ spaces, as well. These theorems are used extensively to prove things in measure theory, functional analysis, and PDEs. The dominated convergence theorem generalizes the result of using uniform convergence to swap limit and integral. We can use these to show that $L^p$ is complete for $pin [1,infty),$ in fact a Banach space, as these define norms. We show that if $p$ is in the range and $X$ is $sigma$-finite, then the dual of $L^p$ is $L^q$, where $1/p+1/q=1,$ and this functional is defined, wait for it, via integration. We're beginning to overlap a bit with functional analysis, so I'll switch gears a bit. We often use the integral to define linear functionals, and one such example is in the Riesz representation theorem. Here, we find that the dual of $C(X)$, where $X$ is a compact metric space, is the space of finite, signed measure of the Borel sigma algebra (Radon measures). In particular, to any bounded linear function $omega$ on $C(X)$, there exists a unique Radon measure $rho$ such that $$omega (f)=intlimits_x fdrho.$$

Also, we get a generalization of the fundamental theorem of calculus using the Hardy-Littlewood maximal function:

Let $fin L^1(mathbb{R}^n, dx)$ and consider $$A_rf(x)=frac{1}{m(B_r)}intlimits_{B_r(x)}f(y)dy,$$ where $r>0.$ Then, $$lim_{rrightarrow 0} A_rf(x)=f(x)$$ a.e.

In fact, if $fin L^p$, then $$lim_{rrightarrow 0}frac{1}{m(B_r)}intlimits_{B_r(x)}|f(y)-f(x)|^p dy=0$$ for a.e. $x$. There's more, but I'd like to move on to functional analysis next. I will edit my post to add these details in later today! I also forgot Fubini/Tonelli, so I definitely need to add that in later!

I would say that the central concept of analysis is the concept of limit, specifically a limit of a sequence. Everything that uses concepts built on the concept of limit I would classify as analysis, not algebra. That includes limit of a series, limit of a function, continuity of a function, derivative and Riemann integral. Then the complex analysis emerges from complex algebra when you introduce complex derivative and integrating over curves. Functional analysis also depends on the concepts of continuity and integral, otherwise it would be just algebra of infinitely-dimensional spaces.

From my viewpoint, Real Analysis is a study of functions (of one or several) real variable. Everything else (limits, derivatives, integrals, infinite series, etc.) is a tool serving this purpose. [There is a mild exception one has to make here for sequences and series of real numbers/vectors; these are functions defined on the set of natural numbers and sometimes, integers.] The theory of real numbers was developed (in the 2nd half of the 19th century) in order to make study of functions rigorous.

For instance, what is the purpose (or, rather, purposes) of computing derivatives of functions? It is to determine if the given function is increasing/decreasing/concave/convex or to approximate the given function by some polynomial (usually a polynomial of degree one).

What is the purpose of computing limits? It is to determine "approximate" behavior of the function when the input variable is close to some (finite or infinite) value.

What is the purpose of computing integrals? It is to compute length (of curves), areas (of surfaces), volumes (of solids), or to find solutions of differential equations (which are equations on functions involving some derivatives). In the geometric problems (lengths, areas and volumes) one computes a single number "measuring" the given function (say, the length of a curve).

What is the purpose of computing Taylor (Fourier) series? It is to approximate functions with polynomials (or sums of trigonometric functions) which are (usually) easier to analyze than general smooth functions.

This is how it was from the very beginning of Real Analysis (Newton, Leibnitz, Bernoulli, Euler and many others).

Mathematical analysis is a mental edifice built up to describe and understand phenomena of geometry, physics, and technics in terms of formulas involving finite mathematical expressions. The core of this all is the study of functions $f:>{mathbb R}to{mathbb R}$ and their properties.

My impression is that Analysis is largely in contrast to Finite/Discrete Math, and thus deals with continuous spaces, especially the real line. This is generalized to spaces with a metric, measure, and/or topology.